Daily Math · 2026-07-18

Question 1

The number of two-digit numbers with distinct digits formable from $\{1, 2, 3, 4, 5\}$ is:

Accepted Answer

$5 \cdot 4 = 20$, which is $A_5^2$.

Answer

$10$

Answer

$20$

Answer

$25$

Answer

$32$

Question 2

For $I_n = \displaystyle\int_0^1 x^n \, dx$, $\lim_{n 	o \infty} I_n$ equals:

Accepted Answer

$\lim \dfrac{1}{n + 1} = 0$.

Answer

$0$

Answer

$1$

Answer

$\infty$

Answer

divergent

Question 3

The complex number $\dfrac{1 + i}{1 - i}$ equals:

Accepted Answer

$\dfrac{1 + i}{1 - i} \cdot \dfrac{1 + i}{1 + i} = \dfrac{(1 + i)^2}{1 - i^2} = \dfrac{2i}{2} = i$.

Answer

$0$

Answer

$1$

Answer

$i$

Answer

$-i$

Question 4

$\mathbb{Z}_6$ is NOT a field because:

Accepted Answer

$\hat{2} \cdot \hat{3} = \hat{6} = \hat{0}$ with $\hat{2}, \hat{3} 
e \hat{0}$ — zero divisors. A field has no zero divisors, so $\mathbb{Z}_6$ is not a field. (Equivalently, $6$ is composite.)

Answer

it is not a ring

Answer

it has no identity

Answer

it has zero divisors (e.g., $\hat{2} \cdot \hat{3} = \hat{0}$)

Answer

it is finite

Question 5

On $\mathcal{P}(X)$ with symmetric difference, the inverse of any set $A$ is:

Accepted Answer

Every element is its own inverse: $A 	riangle A = \emptyset$.

Answer

$A$ itself

Answer

$X \setminus A$

Answer

$\emptyset$

Answer

$X$

Question 6

For the geometric progression $(b_n)_{n \ge 1}$ with $b_1 = 2$ and $q = \sqrt{5}$, compute $\left\lfloor b_4 \right\rfloor$ (the integer part of $b_4$).

Accepted Answer

$b_4 = 10\sqrt{5}$. Since $\sqrt{5} \approx 2.236$, $b_4 \approx 22.36$, so $\lfloor b_4 \rfloor = 22$.

Answer

$20$

Answer

$21$

Answer

$22$

Answer

$25$

Question 7

For which value of $m\in\mathbb{R}$ does the system $\begin{cases}x+y+z=1\ x+my+z=2\ x+y+mz=3\end{cases}$ have **no** solution?

Accepted Answer

$\bullet$ Compute the determinant of the coefficient matrix:

$$\det\begin{pmatrix}1&1&1\1&m&1\1&1&m\end{pmatrix}=1(m^2-1)-1(m-1)+1(1-m)$$

$\bullet$ Simplify:

$$\det = m^2-2m+1 = (m-1)^2$$

$\bullet$ Determinant vanishes only at $m=1$. At $m=1$ all three rows of the coefficient matrix become $(1,1,1)$ but the right-hand side is $(1,2,3)^T$:

$$x+y+z = 1,\ 2,\ 3$$

$\bullet$ Three identical equations with different constants are inconsistent, so the system has no solution iff $m=1$.

Answer

$m=1$

Answer

$m=0$

Answer

$m=-1$

Answer

$m=2$

Answer

$m=-2$

Answer

$m\in\{1,\,-2\}$

Question 8

$\displaystyle\lim_{x 	o 0} \dfrac{\sin 3x}{	an 5x}$ equals:

Accepted Answer

$\dfrac{\sin 3x}{	an 5x} = \dfrac{\sin 3x}{3x} \cdot \dfrac{5x}{	an 5x} \cdot \dfrac{3}{5} 	o 1 \cdot 1 \cdot \dfrac{3}{5} = \dfrac{3}{5}$.

Answer

$	frac{3}{5}$

Answer

$	frac{5}{3}$

Answer

$1$

Answer

$0$

Question 9

How many solutions does $2\sin^2 x - 3\sin x + 1 = 0$ have on $[0, 2\pi)$?

Accepted Answer

$(2\sin x - 1)(\sin x - 1) = 0$. From $\sin x = 	frac{1}{2}$: $x = 	frac{\pi}{6}, 	frac{5\pi}{6}$. From $\sin x = 1$: $x = 	frac{\pi}{2}$. Total: $3$.

Answer

$2$

Answer

$3$

Answer

$4$

Answer

$5$

Question 10

How many real solutions does the equation $|x^2-4| = x+2$ have?

Accepted Answer

$\bullet$ RHS must be non-negative, so $x\geq -2$.

$\bullet$ **Case 1** ($|x|\geq 2$, so $x^2-4\geq 0$):

$$x^2-4=x+2 \Rightarrow x^2-x-6=0$$

$\bullet$ Factor and solve:

$$(x-3)(x+2)=0 \Rightarrow x=3,\ -2$$

$\bullet$ **Case 2** ($|x|<2$, so $x^2-4<0$):

$$4-x^2=x+2 \Rightarrow x^2+x-2=0$$

$\bullet$ Factor and solve:

$$(x+2)(x-1)=0 \Rightarrow x=1$$

(only $x=1$ lies strictly inside $|x|<2$; $x=-2$ already appears in Case 1).

$\bullet$ Distinct valid solutions:

$$\{-2,\ 1,\ 3\}$$

$\bullet$ Three real solutions.

Answer

$1$

Answer

$2$

Answer

$3$

Answer

$4$

Answer

$0$

Answer

$5$

Problems & worked solutions

Problem 1 — Permutations & Combinations

Problem 2 — Recursive Integrals

Problem 3 — Complex Numbers

Problem 4 — Rings & Fields

Problem 5 — Binary Operations

Problem 6 — Geometric Sequences

Problem 7 — Matrices

Problem 8 — Calculus

Problem 9 — Trigonometric Equations

Problem 10 — Algebra